Earlier I said that the projection of the state vector onto one of the
perpendicular axes of its Hilbert Space shows the contribution of that
axes' eigenstate to the whole state. You may wonder what is meant by
the whole state. You would think (and rightly so, according to
classical physics) that our spin-1/2 particle could only exist
entirely in one of the possible +1/2 and -1/2 states, and
accordingly that its state vector could only exist lying completely
along one of its coordinate axes. If the particle's axes are called
x and y, and the state vector's x coordinate which denotes the
contribution to the -1/2, or 0 state, and y coordinate which denotes
the contribution to the +1/2, or 1 state, should only be (1,0), or
(0,1).

That seems perfectly reasonable, but it simply is not correct.
According to quantum physics a quantum system can exist in a mix of
all of its allowed states simultaneously. This is the Principle of
Superposition, and it is key to the power of the quantum computer.
While the physics of superposition is not simple at all,
mathematically it is not difficult to characterize this kind of
behavior.